{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,28]],"date-time":"2026-06-28T07:40:38Z","timestamp":1782632438224,"version":"3.54.5"},"reference-count":52,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,6,10]],"date-time":"2023-06-10T00:00:00Z","timestamp":1686355200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["51975038"],"award-info":[{"award-number":["51975038"]}]},{"name":"National Natural Science Foundation of China","award":["52205083"],"award-info":[{"award-number":["52205083"]}]},{"name":"National Natural Science Foundation of China","award":["52272385"],"award-info":[{"award-number":["52272385"]}]},{"name":"National Natural Science Foundation of China","award":["L211008"],"award-info":[{"award-number":["L211008"]}]},{"name":"National Natural Science Foundation of China","award":["X21055"],"award-info":[{"award-number":["X21055"]}]},{"name":"the Beijing Natural Science Foundation","award":["51975038"],"award-info":[{"award-number":["51975038"]}]},{"name":"the Beijing Natural Science Foundation","award":["52205083"],"award-info":[{"award-number":["52205083"]}]},{"name":"the Beijing Natural Science Foundation","award":["52272385"],"award-info":[{"award-number":["52272385"]}]},{"name":"the Beijing Natural Science Foundation","award":["L211008"],"award-info":[{"award-number":["L211008"]}]},{"name":"the Beijing Natural Science Foundation","award":["X21055"],"award-info":[{"award-number":["X21055"]}]},{"name":"the Fundamental Research Funds for the Beijing Universities","award":["51975038"],"award-info":[{"award-number":["51975038"]}]},{"name":"the Fundamental Research Funds for the Beijing Universities","award":["52205083"],"award-info":[{"award-number":["52205083"]}]},{"name":"the Fundamental Research Funds for the Beijing Universities","award":["52272385"],"award-info":[{"award-number":["52272385"]}]},{"name":"the Fundamental Research Funds for the Beijing Universities","award":["L211008"],"award-info":[{"award-number":["L211008"]}]},{"name":"the Fundamental Research Funds for the Beijing Universities","award":["X21055"],"award-info":[{"award-number":["X21055"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The rapid development of cities in recent years has increased the operational pressure of rail vehicles, and due to the characteristics of rail vehicles, including harsh operating environment, frequent starting and braking, resulting in rails and wheels being prone to rail corrugation, polygons, flat scars and other faults. These faults are coupled in actual operation, leading to the deterioration of the wheel\u2013rail contact relationship and causing harm to driving safety. Hence, the accurate detection of wheel\u2013rail coupled faults will improve the safety of rail vehicles\u2019 operation. The dynamic modeling of rail vehicles is carried out to establish the character models of wheel\u2013rail faults including rail corrugation, polygonization and flat scars to explore the coupling relationship and characteristics under variable speed conditions and to obtain the vertical acceleration of the axle box. An APDM time\u2013frequency analysis method is proposed in this paper based on the PDMF adopting R\u00e9nyi entropy as the evaluation index and employing a WOA to optimize the parameter set. The number of iterations of the WOA adopted in this paper is decreased by 26% and 23%, respectively, compared with PSO and SSA, which means that the WOA performs at faster convergence speed and with a more accurate R\u00e9nyi entropy value. Additionally, TFR obtained using APDM realizes the localization and extraction of the coupled fault characteristics under rail vehicles\u2019 variable speed working conditions with higher energy concentration and stronger noise resistance corresponding to prominent ability of fault diagnosis. Finally, the effectiveness of the proposed method is verified using simulation and experimental results that prove the engineering application value of the proposed method.<\/jats:p>","DOI":"10.3390\/s23125486","type":"journal-article","created":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T02:28:42Z","timestamp":1686536922000},"page":"5486","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Adaptive Parameterized Domain Mapping Method and Its Application in Wheel\u2013Rail Coupled Fault Diagnosis for Rail Vehicles"],"prefix":"10.3390","volume":"23","author":[{"given":"Zihang","family":"Xu","sequence":"first","affiliation":[{"name":"Beijing Key Laboratory of Performance Guarantee on Urban Rail Transit Vehicles, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"},{"name":"School of Mechanical-Electronic and Vehicle Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianwei","family":"Yang","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of Performance Guarantee on Urban Rail Transit Vehicles, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"},{"name":"School of Mechanical-Electronic and Vehicle Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dechen","family":"Yao","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of Performance Guarantee on Urban Rail Transit Vehicles, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"},{"name":"School of Mechanical-Electronic and Vehicle Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0562-3998","authenticated-orcid":false,"given":"Jinhai","family":"Wang","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of Performance Guarantee on Urban Rail Transit Vehicles, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"},{"name":"School of Mechanical-Electronic and Vehicle Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Minghui","family":"Wei","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of Performance Guarantee on Urban Rail Transit Vehicles, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"},{"name":"School of Mechanical-Electronic and Vehicle Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1243\/PIME_PROC_1992_206_226_02","article-title":"An investigation of short pitch wheel and rail corrugations on the Vancouver mass transit system","volume":"206","author":"Kalousek","year":"1992","journal-title":"Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1076\/vesd.40.1.3.15874","article-title":"Train-Track Interaction and Mechanisms of Irregular Wear on Wheel and Rail Surfaces","volume":"40","author":"Nielsen","year":"2003","journal-title":"Veh. Syst. Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s004190050208","article-title":"Polygonalization of railway wheels","volume":"69","author":"Meywerk","year":"1999","journal-title":"Arch. Appl. Mech."},{"key":"ref_4","unstructured":"Johansson, A. (2005). Out-of-Round Railway Wheels\u2014Causes and Consequences an Investigation Including Field Tests, Out-of-Roundness Measurements and Numerical Simulations. [Ph.D. Thesis, Chalmers Tekniska Hogskola]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1817","DOI":"10.1080\/00423114.2012.695022","article-title":"An investigation into the mechanism of the polygonal wear of metro train wheels and its effect on the dynamic behaviour of a wheel\/rail system","volume":"50","author":"Jin","year":"2012","journal-title":"Veh. Syst. Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0043-1648(91)90020-U","article-title":"Factors of wheel flats occurrence and preventive measures","volume":"144","author":"Kumagai","year":"1991","journal-title":"Wear"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1243\/0954409011531459","article-title":"Dynamic effects of vehicles on tracks in the case of raising train speeds","volume":"215","author":"Zhai","year":"2005","journal-title":"Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1177\/09544097221107985","article-title":"Dynamic diagnosis method and quantitative characterization of rail corrugation","volume":"237","author":"Xu","year":"2022","journal-title":"Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1299\/kikaic.66.615","article-title":"Study on Detecting Method of Railway Corrugation Using Wavelet Analysis","volume":"66","author":"Suda","year":"2000","journal-title":"Trans. Jpn. Soc. Mech. Eng."},{"key":"ref_10","first-page":"9","article-title":"Wheel Polygon Fault Diagnosis Based on the Time and requency Domain Characteristics of Axle Box Acceleration","volume":"48","author":"Chen","year":"2021","journal-title":"Machinery"},{"key":"ref_11","first-page":"1","article-title":"Research on Recognition of Wheel Flat Scars Using Wheel-Rail Vibration Acceleration Signal","volume":"48","author":"Yang","year":"2023","journal-title":"Mech. Sci. Technol. Aerosp. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Xu, X., Cui, X., Xu, J., Wen, X., and Yang, Z. (2022). Study on the Interaction between Wheel Polygon and Rail Corrugation in High-Speed Railways. Materials, 15.","DOI":"10.3390\/ma15248765"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"108515","DOI":"10.1016\/j.ymssp.2021.108515","article-title":"Full-scale test and numerical simulation of wheelset-gear box vibration excited by wheel polygon wear and track irregularity","volume":"167","author":"Liu","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3891","DOI":"10.1109\/TSP.2009.2021713","article-title":"Adaptive Nonlinear System Identification in the Short-Time Fourier Transform Domain","volume":"57","author":"Avargel","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5118","DOI":"10.1109\/TSP.2010.2053028","article-title":"Time-Frequency Representation Based on an Adaptive Short-Time Fourier Transform","volume":"58","author":"Zhong","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1142\/S0219530520400047","article-title":"Analysis of adaptive short-time Fourier transform-based synchrosqueezing transform","volume":"19","author":"Cai","year":"2020","journal-title":"Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Guo, Y., Fang, Z., and Chen, X. (2014, January 20\u201323). A new improved Synchrosqueezing Transform based on adaptive short time fourier transform. Proceedings of the 2014 IEEE Far East Forum on Nondestructive Evaluation\/Testing, Chengdu, China.","DOI":"10.1109\/FENDT.2014.6928290"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wang, X., Ying, T., and Tian, W. (2020, January 17\u201319). Spectrum Representation Based on STFT. Proceedings of the 2020 13th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), Chengdu, China.","DOI":"10.1109\/CISP-BMEI51763.2020.9263516"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"124002","DOI":"10.1088\/1361-6501\/ac8abf","article-title":"Fault diagnosis of bearings in multiple working conditions based on adaptive time-varying parameters short-time Fourier synchronous squeeze transform","volume":"33","author":"Wei","year":"2022","journal-title":"Meas. Sci. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ymssp.2015.08.023","article-title":"Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review","volume":"70\u201371","author":"Chen","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3525509","DOI":"10.1109\/TIM.2022.3210987","article-title":"Sparse Time-Frequency Analysis Based on Instantaneous Frequency Estimation and Fault Diagnosis Application","volume":"71","author":"Huang","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Hu, Z., Yang, J., Yao, D., Wang, J., and Bai, Y. (2021). Subway Gearbox Fault Diagnosis Algorithm Based on Adaptive Spline Impact Suppression. Entropy, 23.","DOI":"10.3390\/e23060660"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/TSP.2023.3244105","article-title":"Synchrosqueezed Fractional Wavelet Transform: A New High-Resolution Time-Frequency Representation","volume":"71","author":"Shi","year":"2023","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.ymssp.2018.07.039","article-title":"Parameterised time-frequency analysis methods and their engineering applications: A review of recent advances","volume":"119","author":"Yang","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_25","first-page":"2159","article-title":"A generalized demodulation approach to time-frequency projections for multicomponent signals","volume":"461","author":"Olhede","year":"2005","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2819","DOI":"10.1109\/TIM.2018.2868519","article-title":"Time-Frequency Squeezing and Generalized Demodulation Combined for Variable Speed Bearing Fault Diagnosis","volume":"68","author":"Huang","year":"2019","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"20687","DOI":"10.1109\/ACCESS.2020.2969461","article-title":"An Adaptive Generalized Demodulation Method for Multimedia Spectrum Analysis is Applied in Rolling Bearing Fault Diagnosis","volume":"8","author":"Ma","year":"2020","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2717","DOI":"10.1109\/TII.2022.3192597","article-title":"Flexible Generalized Demodulation for Intelligent Bearing Fault Diagnosis Under Nonstationary Conditions","volume":"19","author":"Liu","year":"2023","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1016\/j.ymssp.2004.05.001","article-title":"Use of the acceleration signal of a gearbox in order to perform angular resampling (with limited speed fluctuation)","volume":"19","author":"Bonnardot","year":"2005","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.mechmachtheory.2018.06.016","article-title":"A modified tacho-less order tracking method for the surveillance and diagnosis of machine under sharp speed variation","volume":"128","author":"Wu","year":"2018","journal-title":"Mech. Mach. Theory"},{"key":"ref_31","first-page":"144","article-title":"Tacho-less Order Tracking for Gearbox based on Time-varying Filter and CWT","volume":"43","author":"Jing","year":"2019","journal-title":"J. Mech. Transm."},{"key":"ref_32","first-page":"266","article-title":"Tacho-less order tracking method of wind turbine gearbox under time-varying conditions","volume":"36","author":"Wan","year":"2023","journal-title":"J. Vib. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.1109\/TSP.2014.2314061","article-title":"General Parameterized Time-Frequency Transform","volume":"62","author":"Yang","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.jsv.2014.09.025","article-title":"Vibration signal analysis using parameterized time\u2013frequency method for features extraction of varying-speed rotary machinery","volume":"335","author":"Yang","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"051014","DOI":"10.1115\/1.4033717","article-title":"Parametric Identification of Nonlinear Vibration Systems Via Polynomial Chirplet Transform","volume":"138","author":"Deng","year":"2016","journal-title":"J. Vib. Acoust."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1370","DOI":"10.1109\/TIE.2016.2612174","article-title":"Chirplet Path Fusion for the Analysis of Time-Varying Frequency-Modulated Signals","volume":"64","author":"Chen","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"101005","DOI":"10.1115\/1.4036993","article-title":"Nonlinear Squeezing Time-Frequency Transform and Application in Rotor Rub-Impact Fault Diagnosis","volume":"139","author":"Wang","year":"2017","journal-title":"J. Manuf. Sci. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1007\/s11465-017-0443-0","article-title":"Non-stationary signal analysis based on general parameterized time\u2013frequency transform and its application in the feature extraction of a rotary machine","volume":"13","author":"Zhou","year":"2017","journal-title":"Front. Mech. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"5791","DOI":"10.1109\/TSP.2022.3220027","article-title":"Parameterized Resampling Time-Frequency Transform","volume":"70","author":"Li","year":"2022","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"7406","DOI":"10.1109\/TIE.2022.3201311","article-title":"Parameterized Domain Mapping for Order Tracking of Rotating Machinery","volume":"70","author":"Li","year":"2023","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Yang, Y., Qiu, X.L., and Peng, Z.K. (2017, January 16\u201318). A Comparison Study between Non-parameterized and Parameterized Time-Frequency Representation for Non-stationary Signals. Proceedings of the 2017 International Conference on Sensing, Diagnostics, Prognostics, and Control (SDPC), Shanghai, China.","DOI":"10.1109\/SDPC.2017.114"},{"key":"ref_42","first-page":"6","article-title":"Improved Binary Grey Wolves Optimization Algorithm Combined with Gaussian Distribution","volume":"55","author":"Chen","year":"2019","journal-title":"Comput. Eng. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"107421","DOI":"10.1016\/j.ymssp.2020.107421","article-title":"Image representation of vibration signals and its application in intelligent compound fault diagnosis in railway vehicle wheelset-axlebox assemblies","volume":"152","author":"Bai","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"106396","DOI":"10.1016\/j.knosys.2020.106396","article-title":"Ensemble transfer CNNs driven by multi-channel signals for fault diagnosis of rotating machinery cross working conditions","volume":"207","author":"He","year":"2020","journal-title":"Knowl.-Based Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3524711","DOI":"10.1109\/TIM.2021.3111977","article-title":"Fault Diagnosis of a Rotor-Bearing System Under Variable Rotating Speeds Using Two-Stage Parameter Transfer and Infrared Thermal Images","volume":"70","author":"Shao","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105118","DOI":"10.1109\/ACCESS.2020.3000068","article-title":"Intelligent Diagnosis for Railway Wheel Flat Using Frequency-Domain Gramian Angular Field and Transfer Learning Network","volume":"8","author":"Bai","year":"2020","journal-title":"IEEE Access"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1080\/23248378.2021.2000897","article-title":"Ensemble decision approach with dislocated time\u2013frequency representation and pre-trained CNN for fault diagnosis of railway vehicle gearboxes under variable conditions","volume":"10","author":"Wang","year":"2021","journal-title":"Int. J. Rail Transp."},{"key":"ref_48","first-page":"100","article-title":"Rolling Bearing Fault Diagnosis Method Based on Parameter Optimization of Sparrow Search Algorithm VMD and K-SVD","volume":"42","author":"Chu","year":"2022","journal-title":"Noise Vib. Control"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The Whale Optimization Algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"6801","DOI":"10.1109\/TIT.2014.2357799","article-title":"On the Conditional R\u00e9nyi Entropy","volume":"60","author":"Fehr","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_51","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle swarm optimization. Proceedings of the ICNN\u201995\u2014International Conference on Neural Networks, Perth, WA, Australia."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1080\/21642583.2019.1708830","article-title":"A novel swarm intelligence optimization approach: Sparrow search algorithm","volume":"8","author":"Xue","year":"2020","journal-title":"Syst. Sci. Control Eng."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/12\/5486\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:52:16Z","timestamp":1760125936000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/12\/5486"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,10]]},"references-count":52,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["s23125486"],"URL":"https:\/\/doi.org\/10.3390\/s23125486","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,10]]}}}